Active Flux and Polygons: why a stabilisation and how to implement boundary conditions

Abstract

In a recent paper, the authors developed an Active Flux-type algorithm for polygonal meshes (see \cite{BP_Pampa_VEM}). The main difference between Active Flux methods on polygonal meshes and those on Cartesian or simplex meshes is that the basis functions are not explicitly known. These functions describe the solution using point-value approximations on the polygon boundaries and the average value within the polygons. In \cite{BP_Pampa_VEM}, it was shown that directly transposing the algorithm from simplexes to polygons leads to a mild instability, and a remedy was proposed without a rigorous explanation. In the present paper, we clarify this issue and demonstrate that the solution presented in \cite{BP_Pampa_VEM} is indeed correct. Furthermore, another challenge-the implementation of boundary conditions-was previously solved heuristically. Building on \cite{pampa_dg}, we show how to properly implement these boundary conditions. Finally, the method is illustrated through several examples with hybrid meshes.

Carleman estimates for a nonconservative Timoshenko beam involving Dirichlet-Neumann boundary conditions and application to exact controllability
Carole Louis-Rose, Thomas Nancy, Louis Tebou
Journal of Mathematical Analysis and Applications14 Oct 2026Peer-reviewedLicence not statedAnalysis & PDEsHAL
Classification of boundary blow-ups for the Mumford-Shah functional with Dirichlet boundary conditions
Francesco Deangelis
arXiv27 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
On the spectrum of the biharmonic Steklov operator with Neumann boundary conditions acting on forms of the unit Euclidean ball
Rodolphe Abou Assali
HAL20 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEs Geometry & topologyHALarXiv
The Three-Dimensional Divergence-Curl Problem with Homogeneous Boundary Conditions and Pseudo-Harmonic Fields
A. V. Gorshkov
arXiv14 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv
Existence and multiplicity results for third-order ordinary differential equations with derivative dependence subject to the Riemann–Stieltjes integral boundary conditions
Du Yue, Cui Yujun
Demonstratio Mathematica26 Oct 2026Peer-reviewedCC BYFull textGeneral mathematicsDOAJ

Cite

BibTeX RIS

BibTeX

@misc{abgrall2026active,
  title = {{Active Flux and Polygons: why a stabilisation and how to implement boundary conditions}},
  author = {Remi Abgrall and Yongle Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34761},
  eprint = {2610.34761},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34761},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Active Flux and Polygons: why a stabilisation and how to implement boundary conditions
AU  - Remi Abgrall
AU  - Yongle Liu
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.34761
UR  - https://arxiv.org/abs/2610.34761
AB  - In a recent paper, the authors developed an Active Flux-type algorithm for polygonal meshes (see \cite{BP_Pampa_VEM}). The main difference between Active Flux methods on polygonal meshes and those on Cartesian or simplex meshes is that the basis functions are not explicitly known. These functions describe the solution using point-value approximations on the polygon boundaries and the average value within the polygons. In \cite{BP_Pampa_VEM}, it was shown that directly transposing the algorithm from simplexes to polygons leads to a mild instability, and a remedy was proposed without a rigorous explanation. In the present paper, we clarify this issue and demonstrate that the solution presented in \cite{BP_Pampa_VEM} is indeed correct. Furthermore, another challenge-the implementation of boundary conditions-was previously solved heuristically. Building on \cite{pampa_dg}, we show how to properly implement these boundary conditions. Finally, the method is illustrated through several examples with hybrid meshes.
ER  -

Abstract

Related

Cite